Cyclotomic extensions of number fields

June 23,2003

Indag. Mathem., N.S., 14 (Z), 183-196

Cyclotomic

Extensions

of Number

Fields*

by Henri Cohen, Francisco Diaz y Diaz and Michel Olivier Laboratoire A2X, U.M. R. 5465 du C.N.R.S., 33405 Talence Cedex, France

VniversitP

Bordeaux

I, 351 Cows

de la LibPration,

Communicated by Prof. M.S. Keane at the meeting of December 16,2002

ABSTRACT

Let K be a number field, k’a prime number, b a primitive e-th root of unity and Kz = K(&). In this paper, we first give a detailed description of the discriminant, conductor, different and prime ideal decomposition of the extension f&/K. We apply this to obtain the Galois-module structure of certain finite modules associated to prime ideals above e, and we also give the Galois-module structure of the unit group of K, modulo eth powers. 1. MAIN

RESULTS

Let K be a number field, let e be a prime number, let ct be a primitive &th root of unity, and set K, = K(&). The purpose of this paper is to study in detail the extension J&/K. Our main reason for doing so is in the application to Kummer theory [l], [2]. Although all of the results are proved by using completely standard elementary methods of algebraic number theory, we have not been able to find them collected in the literature in the explicit way we need them.

Notation.

If L is a number field we will denote by ZL the ring of integers of L. If

L/K is an algebraic extension, we denote by NLIK the relative norm from L to K and by N the absolute norm from K to Q when the field K is understood. For any real number X, we write 1x1 for the ceiling of x, in other words the least integer greater or equal to x. * 2000 Mathematics Subject Classification: primary 1lR18, secondary llR29

183

Finally, if M is some module andf’ is a map from M to itself (often an element of a group algebra), we denote by M[,f the kernel of the map 1’. Since for
@

M[T -gj]

,

O< j
where M[r - gj] is the submodule of elements x E A4 such that r(x) = gj x in M. For any prime ideal p of K such that p ] 4, we will denote by e(p) = e(p/!) the absolute ramification index of p over e (it is consistent to set e(p) = 0 if p is not above 9. The purpose of this paper is to prove the following results. Theorem 1.1. (1) Zf p is aprime ideal of K above C with absolute ramtjication index e(P), the ramtjication index of a prime ideal p, of K, above p is given by the formulas

4 e(pz/p) = (d,, e(p)/qz) = (C -? Lt(p)) In particular,

a prime ideal of K is ramified in K,/K ifand only if p is above e and

(e- We(P). (2) With the same hypotheses, we have

e(P)/% e(P) 4PZl(1 - WQ(Ce))= (4, 4P)/qz) = (e - 1,e(P)) (3) We have

NfGIK (4) The dtrerent, discriminant, ing formulas..

184

and conductor of KzIK are given by the follow-

Theorem 1.2. Let &Ibe a prime ideal of K dividing 4 and let P = npzlp p, be the product of theprime ideals of K, above p. Then, $0 < a < b 5 a + e(p)e(p,/p) we have

Nj(C) =

de - 1, e(P)) -b(P) e-i

1 .

For each d ] d,, let K,[d] be the unique subextension [K, : K,[d]] = d, in other words

of K,/K

such that

In particular, K,[l] = K,, K,[d,] = K and [K,[d] : K] = d,/d. For any prime ideal p of K, we will denote by &idsome prime ideal of KJdl above p, and by and g(nd/n) the ramification index, residual degree, and e(pd/p)y f(t)d/@? number of nd over n, so that, in particular, e(pd/p)f(pd/p)g(pd/p) = d,/d. Note, however, that we write p, (instead of pt) for a prime ideal of K, = K[l] above P. Proposition 1.3. Letf P. We have

f(pd’P)=

In particular,

=f(p,/p)

Cf,d(e--

b e th e residual degree of an ideal p, of K, above

I,e(pi/(e-

l,de(p)))’

ifp is unramified in K,/K we have f (pd/P) = f /cf, d).

Theorem 1.4. Let U = U(K,) be the unit group of K2 For any integer j such that 0
with 185

if j > 2, ,j even if,j>3,j odd.

We are very much indebted to H. W. Lenstra for the proof of Theorem 1.3, and to H. W. Lenstra and J. Martinet for the proof of Theorem 1.4. 2. PROOFS

OF THEOREMS

1.1. 1.2 AND

PROPOSITION

1.3

2.1. Proof of Theorem 1.1

We start by a number of preliminary results. Let p be a prime ideal of K such that p ] e, let pz be a prime ideal of K, above p, and let k = ZK/P and kz = z~~/p~ be the corresponding residue fields. We will denote by e(p,/p), f(p,/p), and g(&/p) the ramification index, residual degree, and number of distinct conjugate ideals of pz over p, by D and Z the decomposition and inertia groups of nr over p, and we let Kf and Kf be the corresponding fixed fields, so that [K; : Kj] = 111= e(pz/p), [Kf : Kf] = lDl/lZl = dimk(k,) =f(p,/p), and [Ki/ : K] = g(p,/p). To simplify notation, we will write e instead of e(p,/p) (not to be confused with e(p) = e(p/a)), and we recall that we write G = Gal(K,/K). Lemma 2.1. There exists a unique integer n such that 0 < n -C e with thefollowing

property. dirnk: 7PZ[‘rd7” - g 9 PZ

ifjzn(mode)

= 1:

otherwise.

Proof. Since ]&/PI] = ~(p~)/N(p,) = ni(pz) = Ik,(, we have dimkr(n,/P~) = 1. On the other hand Z is generated by #:I’, hence the k,-vector space p,/pq is the direct sum of its e eigenspaces for the action of 7dzie.The eigenvalues are gjd:/e for j modulo e, so it follows that exactly one of these eigenspaces has dimension one and the others are zero, from which the lemma follows. 0 Lemma 2.2. For any i > 0, the integer n of the above lemma also satisfies

dimkL $$ ‘.

[..&le _ gi41e] = { A

if j zni (mod e) otherwise.

Once again, since ]pi/pf+ ‘1 = IkJ, the total dimension of the eigenspaces is equal to one so exactly one of them is of dimension one and the others are zero. We must prove that (pi/p;’ *)[#z/e - g”idz/‘] is nonzero. By Lemma 2.1 there exists an x E p, \ &it and y E pz such that h/‘(x) = gdz/‘x + y. If we raise this equation to the ith power, we obtain

Proof.

186

TCle(x’)

=

fidzlexi

+

C

cjxi-jyj

=

g”id;/e$

+ y,

llj
for certain integers cj, and clearly yi E n,i+’ . On the other hand i+,(x) = 1 so 0 vPz(xi) = i, hence xi E pi - nf+‘, proving the lemma. Lemma 2.3. Let e,(p) = vpz(1 - &). Then

(1) Zfn is as above, we have ne,(p) = 1 (mod e). (2) We have e4(p)(l - 1) = e(n=/e) = ee(p). Using the map x H x + 1, it is clear that the additive group pi/pj+ ’ is isomorphic to the multiplicative group (1 + pi) (1 + nf+‘) as a k,[G]-module. By definition of e,(p), we have (‘)) is nonzero. On the other hand, #:l”(&) = &“e. Thus the class of&belongs to the eigenspace of 7d;/’ for the eigenvalue file. It follows from Lemma 2.2 that we have ne,(p) E 1 (mod e), p roving (1). Assertion (2) is immediate since Proof.

(e - l)e,(P) = vp,(l -
2.4. We have

e = 4PJP)

= (&

0

ce1) l,e(p))and

n = e,(p)p’ = (ce ~{~~(p)))P’mod

e.

Proof. The second equality of the lemma gives

(a- 1) eqip)(e-

l,e(n))

=e (!-

e(P) l,e(p))

’

On the other hand, since neq(p) = 1 (mod e), e,(p) is coprime to e. We thus have two irreducible expressions for the rational number e,(p)/e, hence E$)~; e(nA/(Y - 1, e(p)) and e = (e - 1)/(-e - 1, e(p)) from which the corollary

Proof of theorem 1.1. Statement (1) is part of the above corollary.

It implies

that

while e(P,P)

=

VP,(~

=

(e-

so (2) follows by identification.

lb~~(l

-Cd

=

Cl-

lk(PJ(l

- WQ~)

,

In addition, it is clear that 187

so (3) follows from (1) and the formula e(pz/plf(p,/p)g(pz/p) = d,. Since C is coprime to d,, the extension K,/K is tamely ramified, so it follows from (1) that for any prime ideal p, of K, above some prime ideal p of K we have +,(%(K,/K)) = e(pz/n) - 1 = d,/(d,,e(p)/qz) - 1, proving the first formula of (4), and the second formula follows from this and (3) by taking norms. Finally, again because of tame ramification, for every $.I1 b(K:/K) we have roving the last formula and finishing the proof of Theorem “P(f(KIK)) = 1,P 1.1. 0 2.2. Proof of Theorem 1.2

We need some additional preliminary results before proving Theorem 1.2. We keep all the above notation, in particular D, I, k, e = e(p2/p), etc. Consider all the prime ideals pi above p in K,, and set

This is a ring, but in general not a field, and it is a k-module. Lemma 2.5. Let M be as above. Then M is afree k[G/Z]-module of rank 1, in other words there exists a normal basis x E M such that the u(x) for o E G/Z form a k-basis of M. Proof. Let &J,,~be a fixed prime ideal above p in K,. Since Gal((ZKz/p,,O)/k)

z D/Z, the normal basis theorem tells us that there exists 7i E &z/P,,0 such that the u(5) for g E D/Z form a k-basis of &/p,,s. Since the Galois group action is transitive, if c E G is such that a(&?& = p,, it is clear that a(5) is a normal basis of Z,/p, over k. Hence, if x is the element of M whose components are all equal to 0 except at p,,O where the component is zi, it is clear that the g(x) for c E G/Z form a k-basis of M, proving the lemma. cl Corollary

2.6. We have

dimk M[T - g’] = { A

~~e~w~s~mod e,

Proof. A system of representatives of G/Z is given by the 7n for a modulo d,/e. By the above lemma, the E~= p(x) for a modulo d,/e form a k-basis of M which clearly satisfies 7(ea) = en+ 1. Hence, if y = C, moddzleyn&,, we have

188

WY,-1

=gjY*

for all a modulo d,/e.

We thus choose yo, and we must have y, = g-jayo. Applying this to a = dZ/e, we deduce that ys = y&/e = g -jdzleyo, hence if yo # 0 we must have g-jdzle = 1, or equivalently j = 0 (mod e). Thus if j E 0 (mod e) the eigenspace is l-dimensional, otherwise it is 0, proving the corollary. •i Lemma 2.7. The additivegroup Mi = n,,,,* p~/p~+’ is afree M-module of rank 1. More precisely, if x E Mi, then x is an M-basis tfand only if all the components of x are nonzero. Proof. Let x = (x0,) and y = (yP,) be two elements of Mi. It is clear that there exists t = ( tp,) E M such that y = tx if and only if yP, = tp,xp,. Thus if some xp, is equal to 0, x cannot be a basis, and conversely if all xpi are nonzero we can take tp, = ypZ/xP, for all p, so x is a basis. 0 Corollary

2.8. Let x E Mi be a nonzero eigenvector for the action of r, so that T(X) = gjx for some j. Then (1) We have j E ni (mod e), where n = e,(P)-’ (mod e) as in Corollary 2.4 above. (2) x is an M-basis for Mi.

Proof. (1). Since T+(X)

= gjdJe x and x # 0, Lemma

2.2 implies

that

j - ni (mod e).

(2). Since x = (xp,) is nonzero, at least one of the components xpZ,Oof x is nonzero. By transitivity of the Galois action, there exists cr E G, hence of the form 7a, such that a($~~,~)= p,, The equality F(x) = g”‘x implies that xp, = $ixp,,, # 0, so all the components of x are nonzero, and we conclude by Lemma 2.7. 0 It is now easy to obtain the following important

preliminary

result.

Theorem 2.9. Keep all the above notation, in particular e = e(p,/p) = (=t?- l)/(e - l,e(P)) ande,(P) = e(p)/(! - l,e(p)). Then we have dimk pyp -$$-I [r - gj] = { i z

~~er~i~;~p)j

(mod e,

Proof. Let x be a nonzero eigenvector for the action of r on Mi, with eigenvalue gi’ . By the above corollary, we have j’ - ni (mod e), and multiplication by x gives a k-isomorphism of M with Mi. Furthermore, if y E M we have

I = gl’.\-r(~*) hence .\-.r E M,jr - g’] if and only if ,r‘ t M[T g’ “:. H) Corollary 2.6, this eigenspace is equal to 0 except for ,j -,j’ :z 0 (mod t~i when it is one-dimensional, and this condition is ,j 3 j’ = ni (mod e). or i E e,(p)j (mod e), finishing the proof of the theorem. Cl Proof of Theorem 1.2. First note that the condition h < u + e(p)e(pz/p) is necessary and sufficient to insure that P”/Ph is an If!-module. Since the order of G is coprime to e, it is well-known that ffp[G]-exact sequences are split (see Lemma 3.1 below), so that dimk(P”/ph)[r

-g’]

=

c

dimk(P’/P”‘)[r

- ~$1 ,

u < i c h

and Theorem 2.9 says that this is equal to the number of i such that CI5 i < h with i =je(p)/(! - l,e(p)) (mod e). If ni(c) is the number of i zje(p)/ (e - l,e(P)) (mod e) with 0 < i < c, we thus have dimk(P”/Pb)[r -gj] = ni(b) - nj(a). On the other hand, let io be the least nonnegative residue modulo eofh(P)l(~l,dP)).Th e integers i such that 0 5 i < c are thus io, io + e, . , io + (m - l)e, with m = [(h - io)/el. It follows that dimk(P”/ph)[r

-g”]

= [(h - io)/el - [(u - io)/el

It is clear that this expression does not change when ia is changed into io + Xe for any integer A, hence we may replace i.0by je(P)/(! - 1, e(p)), proving Theorem 1.2. 0 2.3. Proof of Proposition

1.3

Proof. Let p be the canonical surjection from G onto Gal(&/K). know that NPdP) = ,@(P,IP)), I(P,/P) = dl(PdP)), and Gal(K,/K,[d]) = < rdzld >. Set as above e = e(nZ/P) = (e - l)/(& - 1, e(p)). We have e(Pd/P)

=

Iz@d/P)i

=

II(Pz/Pl ]Ker(p) n Z(&/P)]

= ] < 7d?ld > t < 7dJr > I

Now if a and b divide d,, we have

I < ra > n < T’ > / = I < @‘(a,h)> 1= dz(a, b)/ab . It follows that e(pd/p)

=

ed, de(dz/d, d,/e) = 6

(e - I)/([

- l,e(P))

= (d, (t - l)/(e - l,e(P)))

e-1 = (t - 1, de(p)) ’ proving the first equality. 190

Then we Ker(d =

e-1 = (d(e - l),de(P), t - 1)

For the second equality, the proof is similar, but now using the decomposition group instead of the inertia group. The details are left to the reader. The third equality follows from the first two by an immediate computation. Finally, from Theorem 1.1 we know that @ is unramified in K,/K if and only if (! - 1) 1e(p), and so the formula forf(p,/p) gives the desired result. 0 3. PROOF

OF THEOREM

3.1. Some Abstract

1.4

Algebraic

Results

For the convenience of the reader, we recall here (with proof) some wellknown basic algebraic results. Lemma 3.1. Let G be afinite Abelian group such that t! 4 (GJ.Any exact sequence of Fp[G]-modules is split. Note that this lemma applies to our situation. Proof. Let

O-A-B&C-O be an exact sequence of ffe[G]-modules. Considering this sequence as a sequence of Et-vector spaces, the sequence splits, hence there exists an (Fe-linear map h from C to B such that g o h = 1~. Define

f =+&xhx-’ xt G Since C+ ]G], this definition makes sense. In addition, since we have averaged over G and g o h = lc, it is clear first that f is ffe[G]-linear, and not only [Fe-linear, and second that we still have g of = lc, proving the lemma. 0 Definition 3.2. Let R be a commutative ring with unit and A and B two R-modules ofjinite length. We say that A and B are Jordan-Holder equivalent (abbreviated A ER.JH B) if they have the same length t and tf their composition factors are isomorphic up to permutation. In other words, A NR,JH B if there exist two increasing sequences (composition series) of submodules of A and B respectively

(0) = A0 c Al c . . . c AI = A (0) = B. c B1 c . . . c B, = B

and

of the same length, such that Ai/Ai- i and Bt/Bi- r are simple and nonzero for 1 < i 5 t, and such that there exists a permutation 7rof the indices such that for each i we have Ai/Ai- 1 E Bj/Bj- 1 forj = r(i). Lemma 3.3. Let R be a commutative ring with unit and A and C two R-modules of 191

finite length. Let B he an R-module such that there exists an exact .srqueme of R-modules O+A+B+C+O. Then up to isomorphism of R-modules, the concatenation A and C gives a composition series,for B.

of

composition series jbr

Indeed, up to isomorphism we can identify A with a submodule of B and C with B/A. Once this identification made, we note that submodules of B/A are of the form Bi/A for Bi a submodule of B containing A, hence as claimed a composition series for B is obtained by concatenating a composition Cl series for A with the lift to B of a composition series for C = B/A. Proof.

Lemma 3.4. Let G be afinite Abelian group such that Ql, ]G] and let A and B be two finite lFr[G]-mo d u 1es. The following three assertions are equivalent. (1) The modules A and B are Jordan-Holder equivalent as Z[Gj-modules. (2) The modules A and B are Jordan-Holder equivalent as Fr[G]-modules. (3) The modules A and B are Fr[G’j-isomorphic.

Since A is a (finite) lFe[G]-module, it is in particular a Z[G]-module, and the set of Z[G]-submodules of A is identical to the set of lFe[G]-submodules of A. In particular, for an Fe[G]-module, composition series are identical whether considered as Fe[G]-composition series or as Z[G]-composition series. This immediately implies that (1) and (2) are equivalent. Since trivially (3) implies (2), it remains to show that (2) implies (3). Thus, assume that A and B are JordanHolder equivalent. Let (At) be the composition factors of a composition series for A as an lFp[G]module. Since by Lemma 3.1 any exact sequence of Fe[G]-modules is split, we deduce that Proof.

It follows by induction that A is isomorphic as an lFe[G]-module to the direct sum of its composition factors. Since B is Jordan-Holder equivalent to A, it I7 follows that B is lFe[G]-isomorphic to A, as claimed. Lemma 3.5. Let G be a$nite Abelian group such that el, G and let M be a$nite Z[G]-module. Then

M/Me=5,[G]

M[C]

.

Proof. Consider the following exact sequence of Z[GJ-modules, middle arrow is the map raising to the &th power:

where the

l-M[e]+M&W44/M~-l By Lemma 3.3 applied to the two small exact sequences 1--+A4[@--+M---+ 192

Me--+1 and l-+Me--+A4--+M/Me -1, as Z[G]-modules a composition series for M can be obtained both as the concatenation of a composition series for M[e] with a composition series for Me, and as the concatenation of a composition series for Me with a composition series for M/Me. By uniqueness of the composition series (the Jordan-Holder theorem), it follows by “simplifying” that the composition series for M/Me is, up to permutation of the composition factors, isomorphic to the composition series for M[l]; in other words, M[e] and M/Me are Jordan-Holder equivalent as Z[G]-modules. Since they both are Fe[G]-modules, we conclude by Lemma 3.4 that they are Fe[G]-isomorphic, as claimed. q Lemma 3.6. Let G be afinite Abelian group. Assume that M and N are two Z[G]-

modules ofjinite type which are free over Z, and that MW2qq

N@zQ.

Then (1) for alln 2 1, MInM

~(z/~z)[G],JH

NW

;

(2) for every prime l such that ll, IGI, M/CM

“F&l N/CN

Proof. Let 4 be an isomorphism of Q[G]-modules from N @z Q to M 8~ Q. We consider N as a submodule of N @z Q. We claim that, without loss of generality, we may replace N by &(N) for any integer k. Indeed,

k~(N)~zQ”~(N)~zQ-N~~a since 4 is an isomorphism

proving Since constant module assume mediate: have

and k is invertible in Q. On the other hand,

our claim. 4(N) is a submodule of finite type in M 8,~ Q, there exists a (nonzero) k such that kqS(N) c M, where as before we identify M with a subof M 8~ Q. Thus, by our above claim, replacing N by k+(N) we may that N c M and [M : N] < 00. The proof of the lemma is now imby definition of Jordan-Holder equivalence over (Z/nZ) [G], we clearly

MInN

--(HI,z)[G],JH

M/N

M/nN

“(Z/d)[G],JH

MInM

@ NW

and

Again by definition,

@ nMlnN

~(z/~E)IG],JH

MInM

@ MIN.

as we have already done in the proof of Lemma 3.5, we 193

may “simplify” by the composition factors of M/N so as to obtain part ( I ) of‘ the lemma. Part (2) follows from this and from Lemma 3.4 applied to A = M/PM and B = N/PN. 0 3.2. Proof of Theorem 1.4

Recall that we assume that &$K, otherwise U/U” is simply an If?-vector space of dimension ~1+ r2. We first introduce some notation. Let L/K be an Abelian extension of number fields with Galois group G (in our case we will have L = KJ, and denote by (RI, R2) the signature of L. Let Soo(L) be the set of infinite places of L, of cardinality RI + Rz. The group G acts on S,(L). More precisely, for any place v of K, the subset of S,(L) of places above v is permuted transitively by G. On the other hand, we let U = U(L) be the group of units of L (of rank RI + RZ - 1). Finally, for a ring F (which will be equal to (w, Q, Z or IFI), we write F’.” = FSxCL)/F(l, l;..,

1).

in other words Fsy” is the quotient of F S,(L) by the subspace of vectors having all equal components. Theorem 3.7. Keep the above notation, and denote by log(U) the usual logarith-

mic embedding of U into IwsxCL). (1) We have an IW-module isomorphism log(u)

@z i.8 =R[G] i@’ .

(2) We have a Q[G]-module isomorphism

(3) For any prime ! such that C+ IGI, we have an [Fe[G]-module isomorphism log(u)

‘& Et E log(u)/

log(Ue)

“O+[G]

5;”

Proof. (1) is essentially a restatement

of Dirichlet’s unit theorem. (2) is a restatement of a theorem of Herbrand on units (see [3] for a proof). (3) Thanks to (2), we can apply Lemma 3.6 (2) to M = log(U) and N = ZsVo to obtain the desired result. 0

Proof of Theorem 1.4. Let p = p(Ki) be the group of roots of unity in K,. From the exact sequence 0--7-~--f U--t log(U) -+O of B[G]-modules we obtain the exact sequence of [Fe[G]-modules

o-+p/pe+

u/ u’ --t(log(u))/(log(u’))~o

Indeed, tensoring with Z/C2 is right-exact, hence the only thing to check is the injectivity of the map from ,u/pe to U/ Ue. But if < E /L is of the form < = r) 194

with n E Kz, then necessarily 77E ~1hence < E $ as claimed. Since [Fe[G]-exact sequences split, and since
We now want to find the eigenspaces for the action of r - gj for 0 < j < d=. We first note that since hence $ =< for some k > 1, and since r(
-

8X

=

C

h,iG,i+l

-

v.i

8

C

&,iev,i v,i

=

C(Xv.i-l v,i

-

giXv,i)ev.i

For this to be 0 in the quotient IFS,0we need all the components to be equal, hence there exists ml E IFI such that for all v and i, Xp,i-l - g’xy,i = ml. Since g is invertible in IFe,we can write x,.i - gjX,,i-t = m for another constant m. Assume first that j # 0, so that g-j # 1. The above recursion in i is immediately solved in terms of x,.0 as x,,i = m( 1 - g-j)-1 + g-Q(xv,o - m( 1 - g-‘)-‘) Taking i = d,/e,, and using that $,‘2 = - 1 when dz is even, we obtain

x P.o = m(l - g-j)-1 + (-l)‘jl”(

x,,~ - m(1 - g-j)-‘)

.

Thus, if 2j/eV is even, this is automatically satisfied, so x,,e can be chosen arbitrarily. This happens exactly if v is unramified or if j is even. On the other hand, if 2j/ey is odd, this fixes the value of x,,o (note that C is odd). Whatever the parity of 2j/e,, the x,,i are given in terms of X,,Ofor i > 0. Thus we consider four cases. (1) The case j even, j > 2. Then all the X,,O can be taken arbitrarily, so the dimension of ( U/ Ue) [r - gj] is equal to 11 + r2. (2) The case j odd, j > 3. Then if v is a real place, X,,O is fixed, so the dimension of ( U/ Up) [r - gj] is equal to r2. (3) The case j = 1. Same as j > 3, except that we get the extra contribution coming from
= 0, hence m = 0 since dz/c>vis coprime to P. It follows that the dimensionof (U/U’)[T-~J] isequal toul +rl -~ I. This finishes the proof of Theorem 1.4. El

mdz/e,

REFERENCES [1] [2] [3]

H. Cohen, F. Diaz y Diaz et M. Olivier, DensifP des discriminants des extensions cycliques degripremier , CR. Acad. Sci. Paris 330 (2000), 61-66. H. Cohen, F. Diaz y Diaz et M. Olivier. On the Density afDiscriminants qfcyclic Extensions Prime Degree, J. reine angew. Math. 550 (2002), 169.-209. S. Lang, Algebraic Number Theory (2nd ed.) , Graduate Texts in Math. 110, Springer-Verlag. 1994.

(Received

196

October

2001)

de oj